# Step-by-Step Solution ## 1. Given Data - **Dewatered sludge:** \( 750 \) kg/d (primary + waste) - **Total solids (TS):** \( 22\% \) - **Quicklime used:** \( 300 \) kg/d (as CaO) - **Process:** Post lime stabilization - **Question:** Calculate theoretical temperature increase after lime addition. --- ## 2. **Reaction and Heat Release** ### **Lime Reaction** Quicklime (CaO) reacts with water: \[ \text{CaO} + \text{H}_2\text{O} \rightarrow \text{Ca(OH)}_2 \] - **Heat of reaction (\(\Delta H\)):** \( -65.2 \) kJ/mol CaO --- ## 3. **Calculate Water in Sludge** ### **Mass of Water in Sludge** \[ \text{Water fraction} = 1 - .22 = .78 \] \[ \text{Water mass per day} = 750\,\text{kg/d} \times .78 = 585\,\text{kg/d} \] --- ## 4. **Calculate Moles of CaO Used** \[ \text{Moles of CaO} = \frac{300\,\text{kg/d}}{56.08\,\text{g/mol}} \] \[ = \frac{300,000\,\text{g}}{56.08\,\text{g/mol}} \approx 5,350\,\text{mol/d} \] --- ## 5. **Total Heat Released** \[ Q = \text{Moles CaO} \times |\Delta H| \] \[ Q = 5,350\,\text{mol} \times 65.2\,\text{kJ/mol} = 349,820\,\text{kJ/d} \] --- ## 6. **Estimate Temperature Increase (\(\Delta T\))** Assume all heat is absorbed by water in the sludge. - **Specific heat of water:** \( 4.18\,\text{kJ/kg}^\circ\text{C} \) - **Mass of water:** \( 585\,\text{kg} \) \[ Q = m c \Delta T \] \[ \Delta T = \frac{Q}{m c} = \frac{349,820}{585 \times 4.18} \] \[ = \frac{349,820}{2,444.3} \approx 143^\circ\text{C} \] --- ### **Realistic Adjustment** - Some heat is lost to surroundings and used to heat solids (not just water). - In practice, temperature rise is often in range \( 30^\circ\text{C} \) to \( 60^\circ\text{C} \). - The above is a theoretical maximum. --- ## **Final Answer (Temperature Increase)** \[ \boxed{143^\circ\text{C}} \] **(theoretical; actual increase will be much lower due to heat losses and specific heat of sludge solids)** --- # 7. **Anaerobic Digestion vs. Lime Stabilization** ## **Advantages of Anaerobic Digestion** - **Energy Production:** Generates biogas (methane) that can be used for heat or electricity. - **Lower Chemical Use:** No need for chemical additives like lime. - **Volume Reduction:** Reduces sludge mass and volume. - **Improved Stabilization:** Better pathogen and odor reduction. - **Potential for Class A biosolids:** Easier land application. ## **Disadvantages of Anaerobic Digestion** - **Higher Capital Cost:** Digesters require significant investment. - **Process Complexity:** Requires careful operation and monitoring. - **Longer Retention Time:** Needs larger tanks and more space. - **Sensitive to Operational Upsets:** Foaming, temperature, pH, and loading rate must be controlled. --- ## **Summary Table** | Factor | Lime Stabilization | Anaerobic Digestion | |---------------------|-----------------------------|------------------------------| | Operating Cost | Lower (chemicals needed) | Higher (energy, O&M) | | Capital Cost | Low | High | | Energy Production | None | Yes (biogas) | | Volume Reduction | Minor | Significant | | Odor/Pathogen Ctrl. | Good | Good/Excellent | | Space Required | Low | High | --- ## **Conclusion** - **Switching to anaerobic digestion** yields long-term sustainability and resource recovery but at higher upfront cost and operational complexity. - **Lime stabilization** is simple and reliable for smaller plants, but incurs recurring chemical costs and does not recover energy. --- **Image: Anaerobic Digester Schematic**